Problem: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{p^2 - 64}{p - 8}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = p$ $ b = \sqrt{64} = -8$ So we can rewrite the expression as: $q = \dfrac{({p} {-8})({p} + {8})} {p - 8} $ We can divide the numerator and denominator by $(p - 8)$ on condition that $p \neq 8$ Therefore $q = p + 8; p \neq 8$